11/22/2021

Meet … Enhydra lutris

sea otter

  • Cute! (but are they really!?)
  • Furry! (guess how may hairs per in2? … (hint, humans are born with ~100,000 total)
  • 150,000 hairs / cm2 (>1,000,000 / in2)

Sea otters: Keystone Species

sea otter keystone

(Estes et al. 1974)

Sea otters: Range

sea otter range

  • Entire (littorally) North Pacific

Sea otters: Furriness > Cuteness

fur trade

  • Fur trade (Russian -> British -> American) leads to near extirpation across the entire range.

  • 300,000 in 1740 … < 2,000 in 1900.

  • Displacement and indenturing of Indigeneous fishermen (esp. Aleut)

fur trade

… the rush for the otters’ “soft gold” was a predictable boom and bust cycle, a cautionary example of unsustainable resource use, and a socioeconomic driver of Western—mainly American—involvement in the Pacific region starting in the eighteenth century. (Loshbaugh 2021)

Sea otters: curiously not totemic

fur trade

art by John Livingston

  • Ainu - Esaman
  • Aleut - Chngatux
  • Alutiiq - Arhnaq
  • Tlingit - Yáxwchʼ
  • Haida - Ku
  • Nuu chah nulth - Кwak̕aƛ
  • Siletz - Elakha

Sea otters: But culturally significant

Sea otter reintroduction: Pacific NW

Remnant populations from Aleutian Islands … released in OR, WA, BC and SE-AK 1969 – 1972.

Reintroduction trade Reintroduction trade

Sea otter reintroduction: Washington State …

Successful!

Population ecology is all about …

\(\huge N\)
  • but where? when?

Here! Now! …

\(\huge N_t\)
  • but how many were there?

That many, then (\(\Delta t\) ago)!

\[\Large N_t = N_{t - \Delta t} + \Delta N\]

slight rearrangement:

\[\Large N_{t+1} = N_t + \Delta N\] For now, we just \(\Delta t = 1\), i.e. it’s the discrete unit that we measure population change. VERY TYPICALLY - whether because of biology or field seasons: \[\Delta t = 1\,\, \textrm{year}\].

How does population change?

\(\Large N_{t+1} = N_t + (B - D) + (I - E)\)

Birth

Death

Immigration

Emigration

Assumption 1: No one’s getting on or off the bus

\(\Large N_{t+1} = N_t + B - D\)

Birth

Death

Immigration

Emigration

This is a closed population … and what we will be (mainly) dealing with for the next 3 weeks.

Assumption 2: the important one

The number of Births and Deaths is proportional to N.

\[\Large N_{t+1} = N_t + bN_t - dN_t\] What does that mean?

  • Every female gives birth to the same number of offspring?
  • Every female has the same probability of giving birth?
  • Every female has the same probability of giving birth to the same distribution of offspring?
  • A fixed proportion of all individuals dies?
  • Every individual has the same probability of dying?
  • the distribution of probabilities of dying is constant?

Some math ….

Redefine \(\lambda = b - d:\)

\[N_{t+1} = N_t + r_0 N_t\] \[N_{t+1} = (1 + r_0) N_t\]

\(r_0\) intrinsic growth, i.e. proportion increase per unit time).

\[N_{t+1} = \lambda N_t\]

Cranking this forward

\[N_{t+1} = \lambda(N_t)\] \[N_{t+2} = \lambda(N_{t+1}) = \lambda^2 N_t\] \[N_{t+3} = \lambda^3 N_t\]

Solution:

\[\large N_{t+y} = \lambda^y N_t\]
Geometric (same as Exponential) growth.

Exponential growth can be very very very fast

Discrete Model to Continuous Model

Let’s do some trickery, starting with: \[N_{t+1} = (1 + r_0) N_t\] \[N_{t+1} - N_t = r_0 N_t\] \[N_{t+\Delta t} - N_t = r_{\Delta t} N_t\]

\[\lim_{\Delta t \to 0} {\Delta N \over \Delta t} = \lim_{\Delta t \to 0} {r_{\Delta t} \over \Delta t} N\]

Magically define: \({r_\Delta \over \Delta t} = r\) and rewrite \(\Delta\) as \(d\):

\[\large {dN \over dt} = r N\]

Solving this differential equation …

\({dN \over dt} = r N\) where \(N(t = 0) = N_0\)

Calculate:

\(\begin{align} {1\over N} dN &= r dt \\ \int_{t' = t_0}^t {1 \over N(t)} dN &= \int_{t' = t_0}^t r dt \\ \log(N) &= rt + C_0 \\ N &= e^{rt + C_0} \\ \end{align}\)

Solution:

Plug in: \[N(0) = N_0\\\] \[ N(t) = N_0 e^{rt}\]

Discrete vs. Continuous Modeling

Difference equations \[N_{t+\Delta t} - N_t = \lambda_{\Delta t} N_t\]

think of absolute change

Pros:

  • Reflects (often) biological reproduction patterns, practical sampling schedule (esp. annual)
  • Intuitive

Cons:

  • Depends on discretization timescale
  • Analytically surprisingly difficult to analyze

Differential equations \[{dN \over dt} = r N\]

think of rates (change/time).

Pros

  • Easier “elegant” mathematical analysis
  • Scales nicely

Cons

  • Unbiological
  • Unintuitive

Sea otter data:

Plot Data:

Fit a line

Plot on Log scale:

Fit linear model log-growth

## 
## Call:
## lm(formula = log(count) ~ I(year - 1970), data = WA)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.191084 -0.062944 -0.005104  0.055518  0.231704 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    4.082641   0.073024   55.91   <2e-16 ***
## I(year - 1970) 0.073251   0.002367   30.95   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1094 on 23 degrees of freedom
## Multiple R-squared:  0.9766, Adjusted R-squared:  0.9755 
## F-statistic: 958.1 on 1 and 23 DF,  p-value: < 2.2e-16

A little math:

\[\log(N_i) = \alpha + \beta \, Y_i\] \[N_i = \exp(\alpha) \times \exp(\beta \, Y_i)\] \[N_i = e^\alpha {e^\beta}^{Y_i}\] \[N_i = N_0 \lambda ^ {Y_i}\]

where \(N_0 = e^{\alpha} = e^{4.08} = 59.14\), and \(\lambda = e^{\beta} = e^{0.07325} = 1.076\).

SO … percent rate of growth is about 7.6%!

Plot exponential growth

Exponential Growth and Natural Selection

There is no exception to the rule that every organic being increases at so high a rate, that if not destroyed, the earth would soon be covered by the progeny of a single pair … The elephant is reckoned to be the slowest breeder of all known animals, and I have taken some pains to estimate its probable minimum rate of natural increase: it will be under the mark to assume that it breeds when thirty years old, and goes on breeding till ninety years old, bringing forth three pairs of young in this interval; if this be so, at the end of the fifth century there would be alive fifteen million elephants, descended from the first pair.

Charles Darwin - Origin of Species

darwin

References

  • J. A. Estes, J. F. Palmisano. 1974. Sea otters: Their role in structuring nearshore communities. Science 185, 1058–1060.
  • Smith et al. 2021. Behavioral responses across a mosaic of ecosystem states restructure a sea otter–urchin trophic cascade. PNAS Mar 2021, 118 (11)
  • Loshbaugh S. 2021. Sea Otters and the Maritime Fur Trade. In: Davis R.W., Pagano A.M. (eds) Ethology and Behavioral Ecology of Sea Otters and Polar Bears. Ethology and Behavioral Ecology of Marine Mammals.
  • Gilkinson, A.K., Pearson, H.C., Weltz, F. and Davis, R.W., 2007. Photo‐identification of sea otters using nose scars. The Journal of Wildlife Management, 71(6), pp.2045-2051.